JEE Physics Electromagnetic Induction Notes Set 04

Magnetic Flux and Faraday's Laws

Magnetic flux through a surface is \( \phi = BA\cos\theta \), where θ is the angle between the field and the normal to the surface; its unit is the weber (T m²). A changing flux induces an emf in the circuit. Faraday's law: \( \varepsilon = -N\,d\phi/dt \). The induced current is \( i = \varepsilon/R \) and the charge that flows is \( q = N\Delta\phi/R \), which does not depend on how fast the flux changes.

Lenz's Law

The induced current flows in a direction that opposes the change of flux that produces it. This is conservation of energy: a magnet falling through a copper tube or a ring is slowed down, so its acceleration is less than g, and it reaches a terminal speed in a long tube. When the flux increases, the induced field is opposite to the original field. When it decreases, it is in the same direction.

Motional emf

1. A straight conductor of length ℓ moving with speed v perpendicular to a field B: \( \varepsilon = B\ell v \). Use the effective length perpendicular to both v and B.
2. A rod of length ℓ rotating about one end with angular speed ω in a field perpendicular to its plane: \( \varepsilon = \tfrac12B\omega\ell^2 \). A disc rotating about its axis behaves in the same way.
3. A loop moving in a uniform field has no net emf, as flux does not change. A conductor moving in a non-uniform field (for example a frame near a long wire) has different emfs on its opposite sides.
4. A coil rotating at frequency f in a uniform field gives \( \varepsilon = NBA\omega\sin\omega t \). The current changes direction twice in each revolution.
5. An induced electric field is produced by a changing magnetic field. Its field lines are closed circles, not tied to charges. Inside a region of radius r the emf is \( -\pi r^2\,dB/dt \) for any loop that encloses the field region.

Inductance

1. Self inductance: flux linkage \( N\phi = Li \), induced emf \( \varepsilon = -L\,di/dt \). For a long solenoid \( L = \mu_0n^2Al \), and for a toroid \( L = \mu_0N^2A/2\pi R \).
2. Mutual inductance: \( \varepsilon_2 = -M\,di_1/dt \). For two coils with perfect coupling \( M = \sqrt{L_1L_2} \).
3. Energy stored in an inductor: \( U = \tfrac12Li^2 \).
4. Growth of current in an LR circuit: \( i = i_0(1 - e^{-t/\tau}) \), where \( \tau = L/R \). Decay: \( i = i_0e^{-t/\tau} \). At the instant the circuit is closed, an inductor behaves like an open circuit.

Illustration 1. A coil of inductance 10 H and resistance 5 Ω is connected to a 5 V battery. Find the current 2 s after closing the switch.
Answer:
\( \tau = L/R = 2 \) s and \( i_0 = V/R = 1 \) A.
\( i = 1(1 - e^{-2/2}) = (1 - e^{-1}) \) A.

Transformer

1. Ideal transformer: \( V_s/V_p = N_s/N_p = I_p/I_s \).
2. Efficiency \( \eta = P_{out}/P_{in} \). For 100% efficiency, \( V_pI_p = V_sI_s \).
3. A transformer works only with AC, because it needs a changing flux.

Illustration 2. A transformer with 90% efficiency takes 6 kW at 200 V. The secondary current is 6 A. Find the secondary voltage and the primary current.
Answer:
Output power = 0.9 × 6000 = 5400 W, so \( V_s = 5400/6 = 900 \) V.
Primary current = 6000/200 = 30 A.